What if both roots are the same: 45 - √n
Using Vieta's formulas:
Sum of roots is a = 90 - 2√n
prod of roots is b = 45^2 - 90√n + n
Since a is positive integer, n must be a square in {0^2, 1^2, 2^2, ..., 44^2}
Thus there are only 45 choices for n, and only 45 choices for a.
a can be in {90, 88, 86, ..., 2}
I put these into a spreadsheet and found one solution where a and b are coprime:
(a,b,n) = (2,1,44^2)
x^2 - 2x + 1
root is 45 - √(44^2) = 1
So one value of n is 44^2 = 1936
https://www.desmos.com/calculator/tu4h87cr7h
I have not fully worked out the case where 45 - √n is one of the roots and the other root is something else.
Edited on April 10, 2024, 10:43 am
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Posted by Larry
on 2024-04-10 10:41:14 |
a solution if double root
